Question

If the polars of $(x_1,y_1)$ and $(x_2,y_2)$ with respect to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are at right angles, then:

• A. $\frac{x_1{x}_2}{y_1{y}_2}-\frac{a^4}{b^4}=0$
• B. $\frac{x_1{x}_2}{y_1{y}_2}+\frac{a^4}{b^4}=0$
• C. $\frac{x_1{x}_2}{y_1{y}_2}+\frac{a^2}{b^2}=0$
• D. $\frac{x_1{x}_2}{y_1{y}_2}-\frac{a^2}{b^4}=0$

Answer 1

The correct option is B $\frac{x_1{x}_2}{y_1{y}_2}+\frac{a^4}{b^4}=0$

The equation of tangents with respect to hyperbola with the polars $(x_1,y_1)$ and $(x_2,y_2)$

$\frac{x_1{x}_2}{{\left(a^2\right)}^2}+\frac{y_1{y}_2}{{\left(b^2\right)}^2}=0\Rightarrow \frac{x_1{x}_2}{a^4}=-\frac{y_1{y}_2}{b^4}\Rightarrow \frac{x_1{x}_2}{y_1{y}_2}=-\frac{a^4}{b^4}\Rightarrow \frac{x_1{x}_2}{y_1{y}_2}+\frac{a^4}{b^4}=0$